Mixed finite element methods for quasilinear second order elliptic problems : the <span class="mathjax-formula">$p$</span>-version

M2AN. M

- M

F. A. M ^ILNER M. S ^URI

Mixed finite element methods for quasilinear second order elliptic problems : the p-version

M2AN. Mathematical modelling and numerical analysis - Modéli- sation mathématique et analyse numérique, tome 26, n^o7 (1992), p. 913-931

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p i MATHEMATlCALMOOQUHGAHDHUMEftlCALAHALYStS b i MOOOtSATHWMATHÉMAnOUtn ANALYSE HUMÉRtQUE

(Vol 26, n° 7, 1992, p 913 à 931)

MIXED FINITE ELEMENT METHODS

FOR QUASILINEAR SECOND ORDER EL LIPT IC PROBLEMS : THE p-VERSION {*)

by F. A. MlLNER 0) and M. SURI (2)

Communicated by R SCOTT

Abstract — The p-version of the finite element method is analyzed for quasilinear second order elhptic problems in mixed weak form Approximation properties of the Raviart-Thomas projection are demonstrated and L2-error bounds for the three relevant variables in the mixed

method are denved

Résumé — Nous analysons la version-p de la méthode d'éléments finis mixtes pour des problèmes quasihnéaires elliptiques du second ordre en forme faible mixte Nous démontrons des propriétés d'approximation de la projection de Raviart-Thomas et on dérive des bornes de V erreur dans L2 (O ) pour les trois variables d'intérêt dans la méthode mixte

I. INTRODUCTION

We consider hère the numerical solution of the following boundary-value problein :

(u) = - V. (ö(M)VM + è(M)) + c(w) = 0 in n ,

M = - flf on df2 ,

where fi is a convex polygon with boundary 9i7, Vw> dénotes the gradient of the scalar function w and V . v and div v dénote the divergence of the vector

(*) Manuscript received October 1991

O Department of Mathematics, Purdue University, West Lafayette, IN 47907, and Diparti- mento di Matematica, lia Umversità di Roma, 00133 Rome, Italy The first author was supported in part by National Science Foundation Grant DMS-8813258

(2) Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21228 The second author was supported in part by Air Force Office of Scientific Research Grant AFOSR-89-0252

M2 AN Modélisation mathématique et Analyse numénque 0764-583X/92/07/913/19/$ 3 90 Mathematica! Modelling and Numerical Analysis © AFCET Gauthier-Villars

914 F A . MILNER, M SURI

function v. We shall assume that for r ^ 2 and for each g e Hr~ 1/2(3 fl ) there exists a unique isolated solution u e Hr(I2) of (1.1) (that is, a solution not situated at a bifurcation point). Note that Sobolev's embedding theorem implies then that u e Wr "l ~£* m (f2 ), e > 0, £ <^ 1, which will be needed throughout the paper.

We shall also assume that the coefficients a : Ô x R -> R,

§ : i2 x R -• R2 and c : O xlR->IR are twice continuously differentiable with bounded derivatives through second order, and that a(x, q) 5* a¹ > 0.

The variable x will be omitted as explicit argument of all functions, except when necessary to avoid ambiguity.

For 1 =s= s ^ oo and k any nonnegative integer, we let Wk's(f2)= {ƒ GLs(ü):Da f eLs(H\ \ a \ ^ k}

dénote the Sobolev space endowed with its standard norm

\a\*k

The subscript J? in the norms will be omitted. Let Hk(O) = Wk>2(f2 ) with norm || . \\k = || \\k r In particular, the notation || - ||0 will mean || . \\L2(n)

or H . \iLï{nf F o r 0 ^ r < o o ]etWr's(f2\ Wr>s(df2\ Hr(O), and Hr(df2) dénote the fractional order Sobolev spaces with norms || . ||r s n,

II • II r s e/2' II • II ^r a ^anc* II ' II r a/3' respectively, defined by interpolation [7].

We shall dénote by ( . , . ) the Hilbert inner product in either L²(f2) or L²(ü)² and by ( ., . ) the L²-inner product on the boundary of O. The same notation will be used to indicate the dualities between Wr's(f2) and W^r> ^s(f2 Y and H^s(df2) and H~^s(dn ), respectively. Throughout the paper, C, Q, and K will dénote genene positive constants which need not have the same value in all their occurrences.

Let

V = //(div ; D ) = {v e L2(n f : div v e L2(f2 )} , normed by

and

W =

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis

MIXED FINITE ELEMENT METHODS : THE p-VERSION 915 The mixed finite element method we shall consider seeks simultaneous approximations of the solution of (1.1), w, and of the flux

z=-a(u)¥u~b(u). (1.2)

The mixed weak formulation of (1.1) consists of finding (z, u) e V x W such that

(a(u)ztv)-(u9óïvv)+(P(u)9v)=(g9v.v)9 v

(div z, w) + (c(u\ w) = 0 , w E W, where we have set

a ( « ) = (1.4)

and v is the outward unit normal vector on 9/2. Our mixed finite element method is a discrete form of (1.3).

Let IS be a décomposition of O by parallelograms which will be the

« éléments » E and let £?p q(E) = {polynomials f(x,y) on E, of degree in x and degree =s # in v}, âp(E) = {polynomials of degree =s/? on

; next define, for e ach element E,

and let

Vp x Wp cz V x W

be the Raviart-Thomas-Nedelec space of index p ^ 0 associated with this décomposition [3, 5], given by

Y

= (fi ^(

>)

{/:^

-^l/

= ƒ . v^£f on £ n £', £, E' e *S }

= f]

where vE dénotes the outward unit normal vector along dE, E s TS. It is known [3, 5] that div Vp c Wp, a property we shall exploit later.

We seek ( / , up) e Vp x Wp so that

/ , g ) - («P, divg)+ , g) = <<?, g . »j> , v eVp , (upX w ) - 0 ,

Equations (1.5) define the /?-version of the mixed finite element approxi- mation for (1.3). This version is based on using a fixed mesh and increasing

vol. 26, n' 7, 1992

9 1 6 F A MILNER, M SURI

the degree of the fini te éléments (as opposed to the &-version that keep s the degree fixed and refines the mesh). The ^-version has been analyzed for the linearized version of ( 1.1 ) in terms of the standard variational form in [ 1 ] and in terms of the mixed variational form in [6]. In this paper, we extend the results obtained in [6] for the linear problem to the quasilinear case. We also obtain an improved version of lemma 3.1 of [6] by reducing the regularity assumed there. We restrict our attention to the mixed method, the corre- sponding generalization for the standard finite element method is more straightforward.

Milner [3] described the h-version of this method for the same problem, demons trated the unique solvability (for small h) of the nonlinear algebraic system (1.5) andderived error estimâtes in Ls (f2), 2 ^ s ==£ + oo, for the error in w, and in //(div ; O ) for the error in z. The assumption there was that the solution of (1.1) was in the space H^{2 + e} (O ). In contrast, for the present paper we shall need an extra half derivative, that is, u E H^{5n + S}(f2).

We shall follow very closely the analysis of [3], In order to do so we shall use the L²-projection onto W^p, P^p :L² -• W^py given by

(ppw-w, x) = 0, xeWp, weW, (1.6)

for which the following approximation properties follow by repeating the arguments of [4] in two dimensions and using interpolation from the cases 5 = 2 and s = oo :

\\ppw- w ||o s*z Qp~m + 3/2'Vs\\w\\m , s > 2 , 3/2 - 31 s ^ m , (1.7) if w e Hm(O). We shall also use the Raviart-Thomas projection of V onto Y^p, TT^P : V -+ V^p, [5] for which we shall demonstrate in Section 2 the following approximation property :

|| ^ - ^ | |0^ Ô P1 / 2~r| | H l |r, r > l / 2 , vsHr(nfn V . (1.8) Our proof of (1.8) improves upon the one presented in [6], which imposed extra regularity on v by requiring that r > 1. In contrast, the condition r > 1/2 is optimal (see remark 2.1). We also obtain estimâtes for the approximation properties of np in the W°7S(f2 )-norm.

We shall find very useful the following inverse-type inequalities, the two dimensional form of the ones found in [4] :

x e L\n )nw

(orx£ L

(n f n v

). (i.9)

The plan of the paper is as foiiows : in Section 2 we demonstrate (1.8), m Section 3 we prove that, for/? sufficiently large, (1.5) is uniquely solvable M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numencal Analysis

MIXED FTNITE ELEMENT METHODS : THE p-VERSION 9 1 7 and its solution (f,u^p) converges to (z, u) in V H L^{2 + £}(O)² x L(2 + 4 e ) / e(/2) for any fixed e, 0 <Z e <è 1, and in Section 4 we establish the rate of convergence of the approximation to the exact solution.

IL THE APPROXIMATION PROPERTIES OF <rr^p

We recall that w^p v is given locally (on every element E) by the following relations (2.1) and (2.2) (see [5]) :

{brpv-v].vE, <p)Si = 0 , «0 6 ^ , , , (2.1) where ( . , . ) s, 1 ^ i ^ 4, dénotes the line intégral along each side S, of the element E and ï?p is the set of all polynomials in one variable of degree less than or equal to /?,

(TTPV-V, £)£ = 0, f eyp(E), (2.2) where ( - , - ) £ dénotes the standard L2(£>inner product.

Now, let /? = [— 1, 1] x [— 1, 1] and let { ^ J^{/ > 0} dénote the L²( [ - 1, l])-complete orthogonal Legendre polynomials. For any v sH(div;R\ let

H*,y)=\t fx^.to^öO, I i^^.W^Cy)], (2.3)

L i = 0 ; = 0 i =0 j =o J

and let

r p + i p P p+i 1

«•'s^y)- E ^a.^F.w^cy), j ; j ; ^ / . w f ^ ) .(2.4)

L. =G j =0 i = 0 j = 0 j

It follows from (2.2)-(2.4) that

\bkl = \i i 0 ^ k =s p , 0 ^ Next, we see that (2.1), (2.3)-(2.5) imply that

p+ 1 oo

J è , ,

P / ± l ) = X & , , , / > , ( ± 1 ) , 0 * 1 « p .

7 = P 7 = P

vol. 26, n° 7, 1992

918 R A . MILNER, M. SURI

S i n c e Pt(- 1 ) = ( - 1 )' a n d Pt(l) = 1, ( 2 . 6 ) implies that

00 00

pj = Xa 2' + P W ' aP+hj = £a2 i + p + l , j > 0 * . ƒ * / > ,

00 00 V '

PROPOSITION 2.1 : Ler g e V and let TTP V be its Raviart-Thomas projection in Vp given by (2.1)-(2.2). Then, if v e Hr(f2)2, we have

||2-^2|lo*fiP

"

ll2ll

. r>H2,

where Q > 0 is a constant independent of p and v but depending on r.

Proof : We first assume that O = R and that the décomposition consists of just one element. Then, v G V and wpv e Vp can be given, respectively, by

(2.3) and (2.4).

The following relation is a trivial conséquence of well known properties of the Legendre polynomials,

\v — irFv\\ = y y — — h

^p{2i

p + 1 00 ^ Q^ 00 p + 1 4

+

+ y y ^hi +

r À <

D(2 i)

^ ^ (2i l)(2 1), ( 2 i + l)(2y + 1) ^ , ( 2 /

i = p + 2 j = p+l v / x J ' i =p+\ j ^ p + 2 v

= I + II H H VIII .

Note that III-VIII can be bounded as follows :

a2 (1 + i2 H- i2Y

while

which implies that (see [6])

III + . . . + VIII=£Öp-2 r||i;||2, r ^ O . (2.9)

M2 AN Modélisation mathématique et Analyse numénque Mathematica! Modelhng and Numencal Analysis

MIXED FINITE ELEMENT METHODS • THE p-VERSION 9 1 9 On the other hand, it follows from (2.7) that

and

2JT1

2^T3^

27T1

(2.10)

II = „

f

2 p + l ^

7_=0

2 Ï + 1

2

4

2/7+ 1 ^ 2i + 1 2/? + 3 ^ 2 Ï + 1

OD

Next observe that b o u n d i n g the series Y (c + fc2)"* ( 1 + 2 ^ ; ) u s i n g t h e

f00 ~P+ A'

intégral method for (c + ?2)~s (1 + 2 f ) dt =s= rP2~2s ( £ > ! ) > and J 5—1

using the Cauchy-Schwarz inequality, we see that, for 5 bounded away from 1,

x (1 + 4 / + 2 / 7 )

^ⁿ 2 i + 1 ^

00 i/2 (^1 + /2 + /2

OP²

-

y -^- —

^ 21 + i

2 / + 1 (2.12) vol. 26, n° 7, 1992

9 2 0 F A. MILNER, M SURI

with exactly the same final bound holding for I £ a2l +p + iw I - It follows from (2.10) and (2.12) that, for s bounded away from 1,

In an entirely analogous w a y (replacing a( i / by bl} and reversing the rôles of i and j) w e d e d u c e from (2.11) that, for s b o u n d e d away from 1,

Combining (2.13) and (2.14) results, for s bounded away from 1, in I + I I ^ Ö / ?1-2 5! ! ^2. (2.15) Next note that

»

( x , ± l ) = J J è , /

T h e trace t h e o r e m ( S o b o l e v ' s e m b e d d i n g theorem) implies that vu

L2(BR) for s > 1/2. Consequently, since Pt(l) = 1 and Pt(- l) = ( - w e see from (2.16) that

Y (+ 1 Y a

CX) ,

00

r °° v

L/ = o J

(2.17)

- 0

Let now v e Hm+£ (H )2. We shall prove that

I k l l . (2.18)

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

MIXED FINITE ELEMENT METHODS • THE p-VERSION 9 2 1 In view of (2.8) and (2.9) it is sufficient to prove that I, II < Qp~ 1E\\V\\ \a + g. It follows from (2.10) that

x(2y + ir

[( i

..i)

+( t ( - ^ X

y =0 L \ Ï =P + 2 / \i =p + 2

- O 1 = 0

4- ) ( — 1 ) 0 , . — ) ( - 1 ) al

\t=0 1=0

p I" / oo \ 2 / o o \ 2 (p+1

4»"

y (2/ + i r

y a.. + ( y (— ïy a., ) + ( y a .

^ Li K J ' \ \ L 1>J \ Lu 1>J J \ L^ 1>J

j =0 L \ Ï =0 / \i =0 / \ i =0

4. /"vr iv \

1

+ l?o ⁽ " ^1)fl '")J

, 2 / p + 1 \2l

+ ( £ (-DXy • (2-

)

; = 0 l _ \ t = 0 / \ i =0 /J

Note that the next to last term on the right hand side of (2.19) can be bounded, using the intégral method for

Jo

^-1/2 - £ __ /->/„l - 2

as p -• oo, as follows :

2 P /+ 1û ? ( l + l2+ j2)1 / 2 + 6 0 / y = 0

with an identical bound holding for the last term of (2.19). Combining (2.19) and (2.20) and using Sobolev's embedding theorem yields

I^Qp-2*\\vJ2m + s. (2.21)

vol 26, n" 7, 1992

9 2 2 F. A. MILNER, M. SURI

In an entirely analogous fashion we can see that

I ~ H 1/2 + e '

which together with (2.21) yields (2.18). Using interpolation [7], it follows from (2.15) and (2.18) that, for s bounded away from 1/2,

which together with (2.8) and (2.9) concludes the proof for the case O — R. For the case when £1 is a disjoint union of parallelograms the resuit follows on each element by using affine mappings onto R. The proposition then follows by summing over all the éléments (see [6] for details).

Remark 2.1 : This resuit differs from the one known for the A-version of the finite element method [3, (1.5)],

| | 2 - * - * £ | |0< Ö Ar| | 2 l |r, r > l / 2 . (2.22) The constraint r :> 1/2 (or r ^ 1/2 + e) stems from the fact that, according to the trace theorem, this is the minimal requirement to ensure that v has a trace on the boundary which is an L2-function (not just a distribution). In [6] the corresponding resuit required an additional half derivative o n r ( r > l ) . In contrast, proposition 2.1 assumes the minimum regularity necessary. It is possible, however, that the bound still holds with the exponent of p replaced by — r, as suggested by (2.22).

COROLLARY 2.1 : For s s* 2, r > max {1/2 ; 3/2 - 3As},

^ I I ^ Q p \ \ v \ \ .

'"" M 0 j •*—• * 11 <-~ 11 y

Proof : Let Pp v be the L2-projection Pp xPp :Y ^>VP. Then the following analogue of (1.7) holds :

\ \ \ \ O s ™\\v\\r, s ^ 2 , 3/2 -3/s*r. (2.23)

Also,

I K ' B - 2 | |

, * | | f ' E - s |

+ | | i r ' 2 - C ' 8 |

, . (2-24)

II^Pi; -ppv\\ ^Qp2-4/s\\<7Tpv -Ppv\\

-v\\o+\\7rPv-v\\o). (2.25) Combining (2.23)-(2.25) and using proposition 2.1, we obtain the corollary.

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

MIXED FINITE ELEMENT METHODS : THE p-VERSION 9 2 3

III. SOLVABILITY OF THE DISCRETE PROBLEM

Following [3] we introducé, for p e Wp, the notation

a(p)- a(u) = - au(p)(u- p) = - au(u)(u- p) + auu(p)(u - pf , (3.1) where

au(p) = au(

Jo

u(p + t[u- p])dt , Jo

and

^uu(P)- O*-t) a^iu+ t[p -u])dt ,

Jo

are bounded functions in Ö. Similarly, we write

(u - p ) = - pu{u){u - p ) + Puu(p ){u - pf ,(3.2) and

c ( p ) - C ( M ) = - c^u{p){u- p) = - c^u(u)(u - p) + c^uu(p)(u - p )² ,

(3.3) where yS^H(p ), )8^MW(p ), c^u(p ), and c^MU(p ) are bounded functions in Ö. Also, let

T = «M(w)^+i?u(w), 7 =cu(u). (3.4) With the notation of (3. l)-(3.4), the following error équations follow from (1.3) and (1.5), [3]:

(a(u)[7rP z- / ] , g ) - (divE, Pp u - up) + ([Pp u - up] T, Ü) =

= {q(up, / ) , 2 ) . £ e V ^ , (div | > * z - ƒ ] , w ) + ( r [ i> /' « - wp] , w ) = ( T ? ( MPX W ) , W G W ^ ,

(3.5) w h e r e

q(up, / ) = a (U)[TTP z - z] + [Pp u - u] F +

+ (a - M^)2 [aWtt(^) z + l ™ , ^ ) ] + au(w^)(M - up)(z - / ) , (3.6)

vol. 26, n° 7, 1992

9 2 4 F. A. MILNER, M. SURI

and

V (*/) = y [P^p M - M] + c^m(u^p)(u - u^pf . (3.7) Just as in [3], we let

&:VpxWp-+Vp xW

be given by &((/JL, p)) = (A, K \ (A, K ) being the (unique) solution of the System

( a ( u ) [ 7 Tpz - A ] , g ) - ( d i v g , Pp u - K ) + ( [ Pp u - K ] F , V ) =

= (?(p, /j), g ) , g e V ' , (div [np z — A ], w ) + (y [Pp u — K], w) = (v (p ), w), w e Wp ,

(3.8) where q{p, JA) and 17 (p) are given by (3.6) and (3.7), respectively, replacing up by p and z*7 by JJL . The unique solvability of this (linear) system follows, for p sufficiently large, from [2], since the left hand side of (3.8) corresponds to the mixed method for the operator M :H2(I2) n HQ(O ) -•

L2(f2) given by

Mw — — V . (a (w ) Vw + a (u ) W.T ) + y o> ,

which has a bounded inverse. In fact, note that (1.2), (1.4) and (3.4) give Mw = - V . [a(u) Vw + a(u)w(au(u) z+ /3u(u))] + cM(w) w

= - V. \a(u)Vw-ha(u)w\-^- (-a(u)Vu) + a (u) bu(u)]\ +

L L a

(u) JJ

= - V . [a(w) Vw + (

which shows that M is the linearization of the operator & in (1.1) about the function w, and, thus, it has a bounded inverse since we have assumed that (1.1) admits unique isolated solutions.

The solvability of (1.5) is now equivalent to showing that <P has a fixed point. This will follow from the Brouwer fixed point theorem if we show that

<P maps a bail of Vp x Wp into itself. We shall need the following technical resuit, a p-version of lemma 2.1 of [3]. Let e > 0 be fixed for the rest of the paper, e < 1.

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

MIXED FINITE ELEMENT METHODS : THE p-VERSION 9 2 5 LEMMA 3.1 : Let 2 =s 0 =s 4 - s. Let <o sV9 qe L2(ü )2, and rj e L2(ft ).

If T eWp satisfies

Ua(u)<o9 g ) - ( d i v 2 , r ) + ( r f , g ) = (<gr, g ) , v e Vp 9

I ( d i v # , H > ) + ( y T , <w) = ( 7 7 , w ) , W G Wp ,

, there exists a constant C = C (0, u, a, F, y, O, e) such that, for p sufficiently large, depending upon e,

Proof : We follow the proof of lemma 2.1 of [3]. Let 6' = 9/(0 - 1 ) be the conjugate exponent of 6. For $ G Le'(O) let <(> eW2>e\n) be the (unique) solution of M* <f> = tf/ in O, t// = 0 on 8/2, where M* is the formai adjoint of M, It follows that || $ ||^{2 Q}, =s Q || $ ||^{0 0I}. We then have [3],

(r, if,) = O?, Ö ( M ) V ^ ) + (*?, 7TPa(u)V<f> -a(u)Y<f>) + + (div o> + yr, <f> — Pp <f>)

+ (a(u)(o + r f , a(u)Y<l> - TT^P a(u)Y<t>) + (v* <t> ) + (V> P^p <t> ~ 4* ) • (3.9) Note that Sobolev's embedding theorem implies that

| | | H ^

,. ( 3 . 1 0 )

Next, (1.8) and Sobolev's embedding theorem imply that (q — a (M) <O, 7TPa(u) V<£ — a(u) Y<f> ) ^

and that

JT, a(u)Y<f> - 7rpa(u)Y<f>)^C\\T\\0 0 \\a(u)Y<t> -

On the other hand, (1.7) and Sobolev's embedding theorem lead to ( d i v « , <f> -Pp <t>)^K\\àiw<o\\Qp-l'2{e\\<f>\\2e,, (3.13)

( y r , <f>-Pp<f>) * K \ \ r \ \O t dp -l-2 / 0 \\<f> \\2^$, , (3.14) vol. 26, n° 7, 1992

9 2 6 F A. MILNER, M SURI a n d

(V,<P)+(v,Pp<f>-<f>)^K\\V\\0\\^\\0^K\\v\\0U\\xe,. (3.15) Collecting (3.9)-(3.15) we see that

( r , ^ ) ¹ ' ² ^ ¹ ^

which, for p sufficiently large, yields the desired estimate.

Now let i£p - Vp with the stronger norm || v \\ rP = || v \\ Q 2 + g + || div v \\ Q

and let i^p = Wp with the stronger norm U w H ^ - ||w||0 r where 4 + 2 s

t = . We can prove now the existence of a solution of (1.5).

THEOREM 3.1 : For ô > 0 sufficiently small (dependent on p) and for p sufficiently large, <P map s a bail of radius ô center ed at (TTP Z, PP U) of -Tp x i1Tp into itself.

Proof: Note that Ut + 1/(2 + e) = 1/2. Let

IK^-dU^ ^{0 and}

Let us use lemma 3.1 on (3.8) with r — Pp u — K, w = TTP z — A, q = q(p, JJL\ v = r] ( p ) and 8 = 4 - e. Observe that (1.7)-(1.9) and corollary 2.1 imply that, for r > 1/2, m = r + 1,

+ || V (P)||o -s 3. \p^m~^r\\ z\\^r + p -^m| | « | |^m + ||« - P ||^,⁴ +

- « +3/2-3/» |

l lr+ i;' (3.16)

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MIXED FINITE ELEMENT METHODS : THE p-VERSION 9 2 7 where ü dépends on ||w||^m. Therefore,

x lldiv (TTP z- A ) | l + 8² + p^m-^r] . (3.17)

On the other hand, taking v = ir^p z — A and w = P^p u - K in (3.8), we see that

\\irpz-à\\0>*2[\\Ppu-K\\0+ \\q\\Q + N | |o ] , (3.18) and, taking w = div (TT^P z - A) in the second équation of (3.8) results in

| | | | | | | | . (3.19)

Combining (3.17)-(3.19) yields the relation

which, for p sufficiently large and r = 5/2, implies that

where the constant ü dépends on || u ||7/2. Combining (3.20) with (1.9) we see that

while (1.9), (3.18), (3.16), and (3.20) imply that

I l i r ' z - A l l , * s 2 p2 e l (-2 + e )\ \ i r P z - A l l

II - - l l o . 2 + . V II - - H o ( 3_2 2 )

**2(ps 62 + p-2+e).

Combining (3.19) and (3.22) yields

- ' * - A | | ^ , * J 2 ( pl«2 + / r2 + ' ) . (3.23) We can now combine (3.21) and (3.23) in the bound

l | J " « - * l l ,

, + \\"

i- iWr^SitP

-"* S

+ P-

-"

)- (3-24)

We want to choose p and ô so that llpl~8/4 d2^— and â{ p~ l ~ e/4 ^ — . vol. 26, n° 7, 1992

9 2 8 F. A. MILNER, M. SURI

Let p s= (2MX )4/% so that I = 2 J j p-! "£/4, ^ is not empty. Then, for S e l , (3.24) implies that

- KW^^Ô and | | i r ' z - A | |r, ^ ô , as we needed.

Remark 3.1 : Note that the choice ô = 2 2LX p~1 " e/4 in theorem 3.1 shows (using (1.7) and (1.8)) not only that (1.5) is solvable but also that, for p -> oo, the solution of (1.5), ( / , up\ differs from (z, M) in the %* x

T^^ norm by 0(p~ l~m) at most. We shall need this observation in order to arrive at the correct error estimâtes.

4. THE Z,2-ERROR BOUNDS

Just as in [3], using (3.1)-(3.3) we now rewrite (3.5) in the form {a (u) Z, V) - (div g, r ) + ( T / \ g) = (4f, g) , g e Vp ,

(4.1) (div £, w) + ( y r , w) = (77, w) , W Ê F ,

where £ = z - / , T=P?U-UP, F = au(up) f + Pu(uP), y = cu{tf\

q = (Fp u -u) F, and ?? = (Fp u - M) y. Note that the left hand side of (4.1) corresponds to the mixed method for the operator Af : H2(fï)-+

L2(O) given by

Nvv = — V . (a(u) Vw 4- a(w) wf) + yw .

Therefore, if we show that its formai adjoint, N *, has a bounded inverse L2 -• H2(O) H H\(O)% then lemma 3.1 would apply to (4.1) without any change in the proof. Since we know that M* has a bounded inverse, ail we need to do is to check that the operator norm of M* - Af* can be made arbitrarily small by taking p large enough.

LEMMA 4.1 : There exists a positive integer p0 such that, for all pi*p0, N* has a bounded inverse L2(D) -> H2(O) n H\(n ). (N * dépends on p through y and F),

Proof : Just as in [3], we have

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis

MIXED FINITE ELEMENT METHODS THE p-VERSION 9 2 9

au(u)- au(up)

where a^uu = and /3^UU, and c^m, defined by analogous u — up

relations, are bounded functions in Ö. It follows from remark 3 1 and Sobolev's embedding theorem that

as needed

To conclude, we establish the rate of convergence of (fy up) to

THEOREM 4.1 : Assume that the solution u of'(1.1) is in HV2(f2 ) There is a positive constant Q, independent of p but dependent on |M|7/2 + 2fi, such

that, for p sufficiently large and m s= 7/2,

0 ll«-K

Ho*fiP

-

IH

,

m) ||divC?-/)||

«j2P

-"

ll«IL

Proof In view of remark 3 1 and lemma 4.1, we can use lemma 3.1 on (4 1) with (9=2. Thus,

2 +p -2| | d i v ^ |o +j |?| jo +| h | |o] (4.2) Note that remark 3.1 together with (1.7) lead to the following estimate for r 5= 0, m > 3/2,

l?io

IMIo= \\{P" u - u) f\\ + || (*»'«-«) y

(4 3)

vol 26, n" 7, 1992

9 3 0 F. A MILNER, M SURI

Combining (4.2), (4.3), (3.18), (3.19), (1.7) and (1.8) yields,

which, for p sufficiently large, leads to

\\r\\⁰^Cp^l-^m\\u\\^mi m ^ 2 , (4.4) where the constant C dépends on ||«||7/2. The first part of the theorem is an immédiate conséquence of (1.7) and (4.4). On the other hand, it follows from (1.8), (3.18), (4.3) and (4.4), that

which proves the second part of the theorem.

Finally, we deduce from (3.19), (1.7), (4.3) and (4.4) that jldiv C ? - / ) | |o« jjdiv 2-PPàlV Z L + |div (^"Z

which gives iii).

Remark4.1 : The estimate for the error in zis the best we could hope for in view of (1.8). The estimate for the error in div z is optimal in rate and regularity, while the one for u is probably not sharp in view of (1.7).

REFERENCES

[1] I. BABUSKA, I. N. KATZ and B. SZABO, The p-version of the finite element method, SIAM Num. Anal., 18 (1981), pp. 515-545.

[2] J. DOUGLAS Jr. and J. E. ROBERTS, Global estimâtes for mixed methods for second order elliptic équations, Math. Comp., 44 (1985), pp. 39-52.

[3] F. A. MELNER, Mixed finite element methodsfor quasilinear second order elliptic problems, Math. Comp., 44 (1985), pp. 303-320.

[4] A. QUARTERONI, Some results of Bernstein and Jackson type for polynomial approximation in LP spaces, Jap. J. Appl. Math., 1 (1984), pp. 173-181.

M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis

MIXED FINITE ELEMENT METHODS : THE p-VERSION 9 3 1 [5] P. A. RAVIART and J. M. THOMAS, A mixed finite element methodfor 2nd order elliptic problems, Proceed. Conf. on Mathematical aspects of finite éléments methods, Lecture notes in mathematics, vol. 606, Springer-Verlag, Berlin, 1977, pp. 292-315.

[6] M. SURI, On the stability and convergence of higher order mixed finite element methods for second order elliptic problems, Math. Comp., 54 (1990), pp. 1-19.

[7] H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978.

vol. 26, n° 7, 1992